# A triangle has corners at #(2 , 2 )#, ( 5, 6 )#, and #( 1, 4 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?

##### 1 Answer

Each of the 3 segments have one endpoint at the midpoint of the given points and the other endpoint at

#### Explanation:

Repeating the points

**Midpoints**

**Slopes of segments** (

Now to simplify the work that remains to do, be noted that the line perpendicular to a side meets one of the other sides (or both at the same time when it intercepts a vertex of the triangle). When it meets just one of the two other sides, which side is this? Answer: the LONG side (although that is intuitive it can be proved). Then we need to know the lengths of the sides of the triangle.

=>

So

line 1 perpendicular to AB meets side BC

line 2 perpendicular to BC meets side AB

line 3 perpendicular to AC meets side AB

We need the equations of the lines in which the sides AB and BC lay and the equations of the 3 perpendicular lines

Equation of the line that supports side:

**[a]**

**[b]**

Equation of the line (passing through midpoint) perpendicular to side:

**[1]**

**[2]**

**[3]**

**Finding the interceptions on sides AB and BC**

Combining equations **[b]** and **[1]**

#{y=(x+7)/2#

#{y=(-3x+26.5)/4# =>#(x+7)/2=(-3x+26.5)/4# =>#2x+14=-3x+26.5# =>#5x=12.5# =>#x=2.5#

#-> y=(2.5+7)/2# =>#y=4.75# We've found

#R(2.5,4.75)#

The distance between#M_(AB)# and R is

#d1=sqrt((2.5-3.5)^2+(4.75-4)^2)=sqrt(1+.5625)=1.25#

Combining equations **[a]** and **[2]**

#{y=(4x-2)/3#

#{y=-2x+11# =>#(4x-2)/3=-2x+11# =>#4x-2=-6x+33# =>#10x=35# =>#x=3.5#

#-> y=-2*3.5+11# =>#y=4# We've found

# S(3.5,4)#

We can find the distance between#M_(BC)# and#S# :

#d2=sqrt((3.5-3)^2+(4-5)^2)=sqrt(1.25)=5*sqrt(5)#

Combining the equations **[a]** and **[3]**

#{y=(4x-2)/3#

#{y=(-x+7.5)/2# =>#(4x-2)/3=(-x+7.5)/2# =>#8x-4=-3x+22.5# =>#11x=26.5# =>#x=53/22#

#-> y=(-53/22+15/2)/2=(-53+165)/44# =>#y=28/11# We've found

#T(53/22, 28/11)#

The distance between#M_(CA)# and T is

#d3=sqrt((53/22-3/2)^2+(28/11-3)^2)=sqrt((10/11)^2+(5/11)^2)=sqrt(125)/11=5*sqrt(5)/11#